Construct Non-Hierarchical P/NBD Model for Short Timeframe Synthetic Data

In this workbook we construct the non-hierarchical P/NBD models on the CD-NOW transaction data.

1 Load and Construct Datasets

1.1 Load Short-Timeframe Synthetic Transaction Data

We now want to load the CD-NOW transaction data.

Code

customer_cohortdata_tbl <- read_rds("data/shortsynth_customer_cohort_data_tbl.rds")
customer_cohortdata_tbl |> glimpse()

Rows: 5,000
Columns: 5
$ customer_id     <chr> "SFC202001_0001", "SFC202001_0002", "SFC202001_0003", …
$ cohort_qtr      <chr> "2020 Q1", "2020 Q1", "2020 Q1", "2020 Q1", "2020 Q1",…
$ cohort_ym       <chr> "2020 01", "2020 01", "2020 01", "2020 01", "2020 01",…
$ first_tnx_date  <dttm> 2020-01-01 15:26:03, 2020-01-01 08:00:48, 2020-01-01 …
$ total_tnx_count <int> 4, 5, 1, 1, 6, 1, 5, 5, 1, 2, 3, 11, 1, 5, 1, 1, 16, 1…

Code

customer_transactions_tbl <- read_rds("data/shortsynth_transaction_data_tbl.rds")
customer_transactions_tbl |> glimpse()

Rows: 29,504
Columns: 7
$ customer_id   <fct> SFC202001_0002, SFC202001_0004, SFC202001_0001, SFC20200…
$ tnx_timestamp <dttm> 2020-01-01 08:00:48, 2020-01-01 14:02:22, 2020-01-01 15…
$ tnx_dow       <fct> Wed, Wed, Wed, Wed, Thu, Thu, Thu, Thu, Thu, Thu, Fri, F…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20200101-0001", "T20200101-0002", "T20200101-0003", "T…
$ tnx_amount    <dbl> 120.46, 208.45, 386.80, 111.52, 133.66, 133.52, 26.19, 1…

Code

customer_subset_id <- read_rds("data/shortsynth_customer_subset_ids.rds")
customer_subset_id |> glimpse()

 Factor w/ 5000 levels "SFC202001_0002",..: 2 3 8 10 14 16 17 21 25 27 ...

We re-produce the visualisation of the transaction times we used in previous workbooks.

Code

plot_tbl <- customer_transactions_tbl |>
  group_nest(customer_id, .key = "cust_data") |>
  filter(map_int(cust_data, nrow) > 3) |>
  slice_sample(n = 30) |>
  unnest(cust_data)

ggplot(plot_tbl, aes(x = tnx_timestamp, y = customer_id)) +
  geom_line() +
  geom_point() +
  labs(
      x = "Date",
      y = "Customer ID",
      title = "Visualisation of Customer Transaction Times"
    ) +
  theme(axis.text.y = element_text(size = 10))

1.2 Load Derived Data

Code

obs_fitdata_tbl   <- read_rds("data/shortsynth_obs_fitdata_tbl.rds")
obs_validdata_tbl <- read_rds("data/shortsynth_obs_validdata_tbl.rds")

customer_fit_stats_tbl <- obs_fitdata_tbl |>
  rename(x = tnx_count)

1.3 Load Subset Data

We also want to construct our data subsets for the purposes of speeding up our valuations.

Code

customer_fit_subset_tbl <- obs_fitdata_tbl |>
  filter(customer_id %in% customer_subset_id)

customer_fit_subset_tbl |> glimpse()

Rows: 1,000
Columns: 6
$ customer_id    <fct> SFC202001_0004, SFC202001_0001, SFC202001_0008, SFC2020…
$ first_tnx_date <dttm> 2020-01-01 14:02:22, 2020-01-01 15:26:03, 2020-01-02 1…
$ last_tnx_date  <dttm> 2020-01-01 14:02:22, 2020-02-27 10:26:24, 2020-06-14 2…
$ tnx_count      <dbl> 0, 3, 4, 0, 10, 0, 2, 0, 1, 4, 1, 16, 3, 2, 7, 0, 1, 5,…
$ t_x            <dbl> 0.0000000, 8.1131300, 23.4784803, 0.0000000, 28.6442666…
$ T_cal          <dbl> 104.3450, 104.3367, 104.2137, 104.1611, 104.0805, 104.0…

Code

customer_valid_subset_tbl <- obs_validdata_tbl |>
  filter(customer_id %in% customer_subset_id)

customer_valid_subset_tbl |> glimpse()

Rows: 1,000
Columns: 3
$ customer_id       <fct> SFC202001_0004, SFC202001_0001, SFC202001_0008, SFC2…
$ tnx_count         <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ tnx_last_interval <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, …

We now use these datasets to set the start and end dates for our various validation methods.

Code

dates_lst <- read_rds("data/shortsynth_simulation_dates.rds")

use_fit_start_date <- dates_lst$use_fit_start_date
use_fit_end_date   <- dates_lst$use_fit_end_date

use_valid_start_date <- dates_lst$use_valid_start_date
use_valid_end_date   <- dates_lst$use_valid_end_date

We now split out the transaction data into fit and validation datasets.

Code

customer_fit_transactions_tbl <- customer_transactions_tbl |>
  filter(
    customer_id %in% customer_subset_id,
    tnx_timestamp >= use_fit_start_date,
    tnx_timestamp <= use_fit_end_date
    )
  
customer_fit_transactions_tbl |> glimpse()

Rows: 5,080
Columns: 7
$ customer_id   <fct> SFC202001_0004, SFC202001_0001, SFC202001_0008, SFC20200…
$ tnx_timestamp <dttm> 2020-01-01 14:02:22, 2020-01-01 15:26:03, 2020-01-02 12…
$ tnx_dow       <fct> Wed, Wed, Thu, Thu, Fri, Fri, Fri, Sat, Sat, Sun, Sun, M…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20200101-0002", "T20200101-0003", "T20200102-0004", "T…
$ tnx_amount    <dbl> 208.45, 386.80, 154.71, 80.41, 135.19, 159.46, 229.79, 9…

Code

customer_valid_transactions_tbl <- customer_transactions_tbl |>
  filter(
    customer_id %in% customer_subset_id,
    tnx_timestamp >= use_valid_start_date,
    tnx_timestamp <= use_valid_end_date
    )
  
customer_valid_transactions_tbl |> glimpse()

Rows: 1,680
Columns: 7
$ customer_id   <fct> SFC202109_0029, SFC202112_0135, SFC202011_0128, SFC20211…
$ tnx_timestamp <dttm> 2022-01-01 01:32:25, 2022-01-01 02:51:43, 2022-01-01 03…
$ tnx_dow       <fct> Sat, Sat, Sat, Sat, Sat, Sat, Sat, Sat, Sun, Sun, Sun, S…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20220101-0004", "T20220101-0006", "T20220101-0008", "T…
$ tnx_amount    <dbl> 99.76, 219.92, 315.00, 207.41, 466.45, 168.03, 107.16, 1…

Finally, we want to extract the first transaction for each customer, so we can add this data to assess our models.

Code

customer_initial_tnx_tbl <- customer_fit_transactions_tbl |>
  slice_min(n = 1, order_by = tnx_timestamp, by = customer_id)

customer_initial_tnx_tbl |> glimpse()

Rows: 1,000
Columns: 7
$ customer_id   <fct> SFC202001_0004, SFC202001_0001, SFC202001_0008, SFC20200…
$ tnx_timestamp <dttm> 2020-01-01 14:02:22, 2020-01-01 15:26:03, 2020-01-02 12…
$ tnx_dow       <fct> Wed, Wed, Thu, Thu, Fri, Fri, Fri, Sat, Sat, Sun, Sun, M…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20200101-0002", "T20200101-0003", "T20200102-0004", "T…
$ tnx_amount    <dbl> 208.45, 386.80, 154.71, 80.41, 135.19, 159.46, 229.79, 9…

We now expand out these initial transactions so that we can append them to our simulations.

Code

sim_init_tbl <- customer_initial_tnx_tbl |>
  transmute(
    customer_id,
    draw_id       = list(1:n_sim),
    tnx_timestamp,
    tnx_amount
    ) |>
  unnest(draw_id)

sim_init_tbl |> glimpse()

Rows: 2,000,000
Columns: 4
$ customer_id   <fct> SFC202001_0004, SFC202001_0004, SFC202001_0004, SFC20200…
$ draw_id       <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1…
$ tnx_timestamp <dttm> 2020-01-01 14:02:22, 2020-01-01 14:02:22, 2020-01-01 14…
$ tnx_amount    <dbl> 208.45, 208.45, 208.45, 208.45, 208.45, 208.45, 208.45, …

Before we start on that, we set a few parameters for the workbook to organise our Stan code.

Code

stan_modeldir <- "stan_models"
stan_codedir  <-   "stan_code"

2 Fit First P/NBD Model

We now construct our Stan model and prepare to fit it with our synthetic dataset.

We also want to set a number of overall parameters for this workbook

To start the fit data, we want to use the 1,000 customers. We also need to calculate the summary statistics for the validation period.

2.1 Compile and Fit Stan Model

We now compile this model using CmdStanR.

Code

pnbd_fixed_stanmodel <- cmdstan_model(
  "stan_code/pnbd_fixed.stan",
  include_paths =   stan_codedir,
  pedantic      =           TRUE,
  dir           =  stan_modeldir
  )

We then use this compiled model with our data to produce a fit of the data.

Code

stan_modelname <- "pnbd_short_fixed1"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")

stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 1.00,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_fixed1_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_short_fixed1_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_fixed1_stanfit <- read_rds(stanfit_object_file)
}

pnbd_short_fixed1_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -51318.32 -51317.35 64.17 63.46 -51423.14 -51210.44 1.00      653
 lambda[1]      0.23      0.21  0.12  0.10      0.09      0.45 1.00     2813
 lambda[2]      0.14      0.08  0.17  0.10      0.00      0.49 1.00     1598
 lambda[3]      0.27      0.25  0.14  0.13      0.09      0.55 1.00     2187
 lambda[4]      0.14      0.08  0.18  0.09      0.00      0.48 1.00     1543
 lambda[5]      0.14      0.08  0.16  0.09      0.01      0.49 1.00     2115
 lambda[6]      0.30      0.28  0.13  0.13      0.13      0.53 1.00     2964
 lambda[7]      0.14      0.12  0.11  0.09      0.02      0.36 1.00     2176
 lambda[8]      0.15      0.14  0.07  0.06      0.06      0.29 1.00     2355
 lambda[9]      0.53      0.50  0.24  0.23      0.20      0.97 1.00     2639
 ess_tail
     1083
     1440
      804
     1061
      674
     1236
     1442
     1150
     1403
     1378

 # showing 10 of 9961 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_fixed1_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed1-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

2.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_fixed1_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_short_fixed1_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_fixed1_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

2.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

We first run the assessment data.

Code

pnbd_stanfit <- pnbd_short_fixed1_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_fixed1_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_short_fixed1",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 5010
  )

pnbd_short_fixed1_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_fixed1_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_fixed1_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_fixed1_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_fixed1_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_fixed1_assess_valid_simstats_tbl.rds"

2.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_fixed1_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

2.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_fixed1_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

3 Fit Alternate Prior Model.

We want to try an alternate prior model with a smaller co-efficient of variation to see what impact it has on our procedures.

Code

stan_modelname <- "pnbd_short_fixed2"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 0.50,
    
    mu_mn     = 0.10,
    mu_cv     = 0.50,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_fixed2_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )

  pnbd_short_fixed2_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_fixed2_stanfit <- read_rds(stanfit_object_file)
}

pnbd_short_fixed2_stanfit$print()

  variable       mean     median    sd   mad         q5        q95 rhat
 lp__      -109570.21 -109570.00 58.99 59.30 -109669.05 -109472.95 1.01
 lambda[1]       0.24       0.23  0.09  0.08       0.12       0.40 1.01
 lambda[2]       0.21       0.19  0.11  0.10       0.07       0.42 1.00
 lambda[3]       0.26       0.25  0.10  0.10       0.12       0.44 1.00
 lambda[4]       0.21       0.19  0.12  0.10       0.07       0.44 1.00
 lambda[5]       0.21       0.19  0.12  0.10       0.06       0.42 1.00
 lambda[6]       0.28       0.27  0.10  0.09       0.14       0.46 1.01
 lambda[7]       0.20       0.19  0.09  0.08       0.08       0.36 1.00
 lambda[8]       0.18       0.18  0.07  0.07       0.09       0.30 1.00
 lambda[9]       0.37       0.35  0.13  0.13       0.18       0.61 1.00
 ess_bulk ess_tail
      703     1165
     3951     1302
     3320     1191
     2928     1273
     4116     1338
     3839     1305
     3770     1166
     5187     1307
     3372     1170
     3617     1142

 # showing 10 of 9961 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_fixed2_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed2-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

3.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_fixed2_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Code

pnbd_short_fixed2_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_fixed2_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

3.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

We first run the assessment data.

Code

pnbd_stanfit <- pnbd_short_fixed2_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_fixed2_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_short_fixed2",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 5020
  )

pnbd_short_fixed2_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_fixed2_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_fixed2_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_fixed2_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_fixed2_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_fixed2_assess_valid_simstats_tbl.rds"

3.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_fixed2_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

3.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_fixed2_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

4 Fit Tight-Lifetime Model

We now want to try a model where we use priors with a tighter coefficient of variation for lifetime but keep the CoV for transaction frequency.

Code

stan_modelname <- "pnbd_short_fixed3"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 0.50,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_fixed3_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )

  pnbd_short_fixed3_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_fixed3_stanfit <- read_rds(stanfit_object_file)
}
  
pnbd_short_fixed3_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -84758.37 -84759.20 59.76 60.71 -84857.21 -84663.30 1.01      690
 lambda[1]      0.24      0.22  0.11  0.11      0.09      0.45 1.00     5043
 lambda[2]      0.14      0.08  0.18  0.09      0.01      0.50 1.00     3401
 lambda[3]      0.27      0.25  0.15  0.13      0.09      0.56 1.00     3816
 lambda[4]      0.14      0.08  0.16  0.09      0.00      0.45 1.00     2396
 lambda[5]      0.14      0.08  0.17  0.09      0.01      0.49 1.00     2727
 lambda[6]      0.30      0.28  0.13  0.12      0.12      0.55 1.01     3970
 lambda[7]      0.14      0.11  0.11  0.09      0.02      0.36 1.00     3170
 lambda[8]      0.16      0.14  0.08  0.07      0.06      0.30 1.00     4324
 lambda[9]      0.53      0.49  0.24  0.23      0.20      0.99 1.01     4048
 ess_tail
     1176
     1595
     1512
      909
      885
     1151
     1382
     1190
     1368
     1268

 # showing 10 of 9961 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_fixed3_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed3-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed3-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed3-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed3-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

4.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_fixed3_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Code

pnbd_short_fixed3_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_fixed3_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

4.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

We first run the assessment data.

Code

pnbd_stanfit <- pnbd_short_fixed3_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_fixed3_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_short_fixed3",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 5030
  )

pnbd_short_fixed3_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_fixed3_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_fixed3_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_fixed3_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_fixed3_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_fixed3_assess_valid_simstats_tbl.rds"

4.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_fixed3_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

4.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_fixed3_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

5 Fit Narrow-Short-Lifetime Model

We now want to try a model where we use priors with a tighter coefficient of variation for lifetime but keep the CoV for transaction frequency.

Code

stan_modelname <- "pnbd_short_fixed4"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.20,
    mu_cv     = 0.30,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_fixed4_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )

  pnbd_short_fixed4_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_fixed4_stanfit <- read_rds(stanfit_object_file)
}

pnbd_short_fixed4_stanfit$print()

  variable       mean     median    sd   mad         q5        q95 rhat
 lp__      -138954.82 -138955.00 60.33 60.79 -139052.00 -138855.00 1.00
 lambda[1]       0.25       0.23  0.11  0.11       0.10       0.45 1.00
 lambda[2]       0.16       0.10  0.18  0.11       0.01       0.53 1.00
 lambda[3]       0.28       0.26  0.14  0.13       0.10       0.56 1.00
 lambda[4]       0.16       0.10  0.19  0.11       0.01       0.49 1.00
 lambda[5]       0.16       0.10  0.18  0.11       0.01       0.52 1.00
 lambda[6]       0.30       0.28  0.13  0.12       0.13       0.55 1.00
 lambda[7]       0.16       0.13  0.12  0.09       0.03       0.38 1.00
 lambda[8]       0.16       0.15  0.08  0.07       0.06       0.31 1.00
 lambda[9]       0.55       0.51  0.25  0.23       0.22       1.01 1.00
 ess_bulk ess_tail
      777     1099
     3926     1483
     3148     1179
     3440     1415
     2503     1020
     2253     1047
     4435     1558
     3253      982
     4072     1240
     3946     1443

 # showing 10 of 9961 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_fixed4_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed4-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed4-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed4-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed4-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

5.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_fixed4_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Code

pnbd_short_fixed4_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_fixed4_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

5.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

We first run the assessment data.

Code

pnbd_stanfit <- pnbd_short_fixed4_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_fixed4_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_short_fixed4",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 5040
  )

pnbd_short_fixed4_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_fixed4_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_fixed4_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_fixed4_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_fixed4_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_fixed4_assess_valid_simstats_tbl.rds"

5.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_fixed4_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

5.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_fixed4_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

6 Compare Model Outputs

We have looked at each of the models individually, but it is also worth looking at each of the models as a group.

We now want to combine both the fit and valid transaction sets to calculate the summary statistics for both.

Code

obs_summstats_tbl <- list(
    fit   = customer_fit_transactions_tbl,
    valid = customer_valid_transactions_tbl
    ) |>
  bind_rows(.id = "assess_type") |>
  group_by(assess_type) |>
  calculate_transaction_summary_statistics() |>
  pivot_longer(
    cols      = !assess_type,
    names_to  = "label",
    values_to = "obs_value"
    )

obs_summstats_tbl |> glimpse()

Rows: 16
Columns: 3
$ assess_type <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", "v…
$ label       <chr> "p10", "p25", "p50", "p75", "p90", "p99", "total_count", "…
$ obs_value   <dbl> 1.000000, 1.000000, 2.000000, 5.000000, 12.000000, 45.0000…

Code

model_assess_transactions_tbl <- dir_ls("data", regexp = "pnbd_short_fixed.*_assess_.*index") |>
  enframe(name = NULL, value = "file_path") |>
  mutate(
    model_label = str_replace(file_path, "data/pnbd_short_(.*?)_assess_.*", "\\1"),
    assess_type = if_else(str_detect(file_path, "_assess_fit_"), "fit", "valid"),
    
    assess_data = map(
      file_path, construct_model_assessment_data,
      
      .progress = "construct_assess_data"
      )
    ) |>
  select(model_label, assess_type, assess_data) |>
  unnest(assess_data)

model_assess_transactions_tbl |> glimpse()

Rows: 31,381,370
Columns: 6
$ model_label   <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed1", "fixed…
$ assess_type   <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", …
$ customer_id   <fct> SFC202001_0004, SFC202001_0004, SFC202001_0004, SFC20200…
$ draw_id       <int> 3, 3, 3, 3, 4, 4, 6, 6, 6, 6, 6, 6, 6, 8, 8, 9, 9, 9, 10…
$ tnx_timestamp <dttm> 2020-01-02 21:24:06, 2020-01-10 06:43:55, 2020-01-15 18…
$ tnx_amount    <dbl> 4.41, 186.28, 40.43, 133.27, 52.23, 11.18, 219.15, 252.7…

We now want to calculate the transaction statistics on this full dataset, for each separate draw.

Code

model_assess_tbl <- model_assess_transactions_tbl |>
  group_by(model_label, assess_type, draw_id) |>
  calculate_transaction_summary_statistics()

model_assess_tbl |> glimpse()

Rows: 16,000
Columns: 11
$ model_label <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed1", "fixed1"…
$ assess_type <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", "f…
$ draw_id     <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,…
$ p10         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
$ p25         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1…
$ p50         <dbl> 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0…
$ p75         <dbl> 7, 7, 9, 7, 8, 7, 7, 9, 7, 8, 7, 8, 8, 8, 8, 8, 7, 7, 8, 7…
$ p90         <dbl> 14.0, 17.2, 18.3, 18.0, 20.0, 16.1, 16.3, 21.0, 16.0, 17.0…
$ p99         <dbl> 55.52, 51.06, 53.39, 44.60, 50.18, 45.42, 44.86, 49.25, 45…
$ total_count <int> 3945, 4161, 4391, 4047, 4377, 3860, 3975, 4391, 4135, 4186…
$ mean_count  <dbl> 6.477833, 6.946578, 7.467687, 6.623568, 7.319398, 6.655172…

We now combine all this data to create a number of different comparison plots for the various summary statistics.

Code

#! echo: TRUE

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "total_count", "Total Transactions"
  )

Code

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "mean_count", "Average Transactions per Customer"
  )

Code

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "p99", "99th Percentile Count"
  )

6.1 Write Assessment Data to Disk

We now want to save the assessment data to disk.

Code

model_assess_tbl |> write_rds("data/assess_data_pnbd_short_fixed_tbl.rds")

R Environment

Code

options(width = 120L)
sessioninfo::session_info()

─ Session info ───────────────────────────────────────────────────────────────────────────────────────────────────────
 setting  value
 version  R version 4.3.1 (2023-06-16)
 os       Ubuntu 22.04.3 LTS
 system   x86_64, linux-gnu
 ui       X11
 language (EN)
 collate  en_US.UTF-8
 ctype    en_US.UTF-8
 tz       Europe/Dublin
 date     2023-12-05
 pandoc   3.1.1 @ /usr/local/bin/ (via rmarkdown)

─ Packages ───────────────────────────────────────────────────────────────────────────────────────────────────────────
 package        * version    date (UTC) lib source
 abind            1.4-5      2016-07-21 [1] RSPM (R 4.3.0)
 arrayhelpers     1.1-0      2020-02-04 [1] RSPM (R 4.3.0)
 backports        1.4.1      2021-12-13 [1] RSPM (R 4.3.0)
 base64enc        0.1-3      2015-07-28 [1] RSPM (R 4.3.0)
 bayesplot      * 1.10.0     2022-11-16 [1] RSPM (R 4.3.0)
 bit              4.0.5      2022-11-15 [1] RSPM (R 4.3.0)
 bit64            4.0.5      2020-08-30 [1] RSPM (R 4.3.0)
 bridgesampling   1.1-2      2021-04-16 [1] RSPM (R 4.3.0)
 brms           * 2.20.4     2023-09-25 [1] RSPM (R 4.3.0)
 Brobdingnag      1.2-9      2022-10-19 [1] RSPM (R 4.3.0)
 cachem           1.0.8      2023-05-01 [1] RSPM (R 4.3.0)
 callr            3.7.3      2022-11-02 [1] RSPM (R 4.3.0)
 checkmate        2.3.0      2023-10-25 [1] RSPM (R 4.3.0)
 cli              3.6.1      2023-03-23 [1] RSPM (R 4.3.0)
 cmdstanr       * 0.6.0.9000 2023-11-21 [1] Github (stan-dev/cmdstanr@a13c798)
 coda             0.19-4     2020-09-30 [1] RSPM (R 4.3.0)
 codetools        0.2-19     2023-02-01 [2] CRAN (R 4.3.1)
 colorspace       2.1-0      2023-01-23 [1] RSPM (R 4.3.0)
 colourpicker     1.3.0      2023-08-21 [1] RSPM (R 4.3.0)
 conflicted     * 1.2.0      2023-02-01 [1] RSPM (R 4.3.0)
 cowplot        * 1.1.1      2020-12-30 [1] RSPM (R 4.3.0)
 crayon           1.5.2      2022-09-29 [1] RSPM (R 4.3.0)
 crosstalk        1.2.0      2021-11-04 [1] RSPM (R 4.3.0)
 curl             5.1.0      2023-10-02 [1] RSPM (R 4.3.0)
 digest           0.6.33     2023-07-07 [1] RSPM (R 4.3.0)
 directlabels   * 2023.8.25  2023-09-01 [1] RSPM (R 4.3.0)
 distributional   0.3.2      2023-03-22 [1] RSPM (R 4.3.0)
 dplyr          * 1.1.3      2023-09-03 [1] RSPM (R 4.3.0)
 DT               0.30       2023-10-05 [1] RSPM (R 4.3.0)
 dygraphs         1.1.1.6    2018-07-11 [1] RSPM (R 4.3.0)
 ellipsis         0.3.2      2021-04-29 [1] RSPM (R 4.3.0)
 evaluate         0.22       2023-09-29 [1] RSPM (R 4.3.0)
 fansi            1.0.5      2023-10-08 [1] RSPM (R 4.3.0)
 farver           2.1.1      2022-07-06 [1] RSPM (R 4.3.0)
 fastmap          1.1.1      2023-02-24 [1] RSPM (R 4.3.0)
 forcats        * 1.0.0      2023-01-29 [1] RSPM (R 4.3.0)
 fs             * 1.6.3      2023-07-20 [1] RSPM (R 4.3.0)
 furrr          * 0.3.1      2022-08-15 [1] RSPM (R 4.3.0)
 future         * 1.33.0     2023-07-01 [1] RSPM (R 4.3.0)
 generics         0.1.3      2022-07-05 [1] RSPM (R 4.3.0)
 ggdist           3.3.0      2023-05-13 [1] RSPM (R 4.3.0)
 ggplot2        * 3.4.4      2023-10-12 [1] RSPM (R 4.3.0)
 globals          0.16.2     2022-11-21 [1] RSPM (R 4.3.0)
 glue           * 1.6.2      2022-02-24 [1] RSPM (R 4.3.0)
 gridExtra        2.3        2017-09-09 [1] RSPM (R 4.3.0)
 gtable           0.3.4      2023-08-21 [1] RSPM (R 4.3.0)
 gtools           3.9.4      2022-11-27 [1] RSPM (R 4.3.0)
 hms              1.1.3      2023-03-21 [1] RSPM (R 4.3.0)
 htmltools        0.5.6.1    2023-10-06 [1] RSPM (R 4.3.0)
 htmlwidgets      1.6.2      2023-03-17 [1] RSPM (R 4.3.0)
 httpuv           1.6.12     2023-10-23 [1] RSPM (R 4.3.0)
 igraph           1.5.1      2023-08-10 [1] RSPM (R 4.3.0)
 inline           0.3.19     2021-05-31 [1] RSPM (R 4.3.0)
 jsonlite         1.8.7      2023-06-29 [1] RSPM (R 4.3.0)
 knitr            1.44       2023-09-11 [1] RSPM (R 4.3.0)
 labeling         0.4.3      2023-08-29 [1] RSPM (R 4.3.0)
 later            1.3.1      2023-05-02 [1] RSPM (R 4.3.0)
 lattice          0.21-8     2023-04-05 [2] CRAN (R 4.3.1)
 lifecycle        1.0.3      2022-10-07 [1] RSPM (R 4.3.0)
 listenv          0.9.0      2022-12-16 [1] RSPM (R 4.3.0)
 loo              2.6.0      2023-03-31 [1] RSPM (R 4.3.0)
 lubridate      * 1.9.3      2023-09-27 [1] RSPM (R 4.3.0)
 magrittr       * 2.0.3      2022-03-30 [1] RSPM (R 4.3.0)
 markdown         1.11       2023-10-19 [1] RSPM (R 4.3.0)
 Matrix           1.5-4.1    2023-05-18 [2] CRAN (R 4.3.1)
 matrixStats      1.0.0      2023-06-02 [1] RSPM (R 4.3.0)
 memoise          2.0.1      2021-11-26 [1] RSPM (R 4.3.0)
 mime             0.12       2021-09-28 [1] RSPM (R 4.3.0)
 miniUI           0.1.1.1    2018-05-18 [1] RSPM (R 4.3.0)
 munsell          0.5.0      2018-06-12 [1] RSPM (R 4.3.0)
 mvtnorm          1.2-3      2023-08-25 [1] RSPM (R 4.3.0)
 nlme             3.1-162    2023-01-31 [2] CRAN (R 4.3.1)
 parallelly       1.36.0     2023-05-26 [1] RSPM (R 4.3.0)
 pillar           1.9.0      2023-03-22 [1] RSPM (R 4.3.0)
 pkgbuild         1.4.2      2023-06-26 [1] RSPM (R 4.3.0)
 pkgconfig        2.0.3      2019-09-22 [1] RSPM (R 4.3.0)
 plyr             1.8.9      2023-10-02 [1] RSPM (R 4.3.0)
 posterior      * 1.4.1      2023-03-14 [1] RSPM (R 4.3.0)
 prettyunits      1.2.0      2023-09-24 [1] RSPM (R 4.3.0)
 processx         3.8.2      2023-06-30 [1] RSPM (R 4.3.0)
 promises         1.2.1      2023-08-10 [1] RSPM (R 4.3.0)
 ps               1.7.5      2023-04-18 [1] RSPM (R 4.3.0)
 purrr          * 1.0.2      2023-08-10 [1] RSPM (R 4.3.0)
 quadprog         1.5-8      2019-11-20 [1] RSPM (R 4.3.0)
 QuickJSR         1.0.7      2023-10-15 [1] RSPM (R 4.3.0)
 R6               2.5.1      2021-08-19 [1] RSPM (R 4.3.0)
 Rcpp           * 1.0.11     2023-07-06 [1] RSPM (R 4.3.0)
 RcppParallel     5.1.7      2023-02-27 [1] RSPM (R 4.3.0)
 readr          * 2.1.4      2023-02-10 [1] RSPM (R 4.3.0)
 reshape2         1.4.4      2020-04-09 [1] RSPM (R 4.3.0)
 rlang          * 1.1.1      2023-04-28 [1] RSPM (R 4.3.0)
 rmarkdown        2.25       2023-09-18 [1] RSPM (R 4.3.0)
 rstan            2.32.3     2023-10-15 [1] RSPM (R 4.3.0)
 rstantools       2.3.1.1    2023-07-18 [1] RSPM (R 4.3.0)
 rstudioapi       0.15.0     2023-07-07 [1] RSPM (R 4.3.0)
 scales         * 1.2.1      2022-08-20 [1] RSPM (R 4.3.0)
 sessioninfo      1.2.2      2021-12-06 [1] RSPM (R 4.3.0)
 shiny            1.7.5.1    2023-10-14 [1] RSPM (R 4.3.0)
 shinyjs          2.1.0      2021-12-23 [1] RSPM (R 4.3.0)
 shinystan        2.6.0      2022-03-03 [1] RSPM (R 4.3.0)
 shinythemes      1.2.0      2021-01-25 [1] RSPM (R 4.3.0)
 StanHeaders      2.26.28    2023-09-07 [1] RSPM (R 4.3.0)
 stringi          1.7.12     2023-01-11 [1] RSPM (R 4.3.0)
 stringr        * 1.5.0      2022-12-02 [1] RSPM (R 4.3.0)
 svUnit           1.0.6      2021-04-19 [1] RSPM (R 4.3.0)
 tensorA          0.36.2     2020-11-19 [1] RSPM (R 4.3.0)
 threejs          0.3.3      2020-01-21 [1] RSPM (R 4.3.0)
 tibble         * 3.2.1      2023-03-20 [1] RSPM (R 4.3.0)
 tidybayes      * 3.0.6      2023-08-12 [1] RSPM (R 4.3.0)
 tidyr          * 1.3.0      2023-01-24 [1] RSPM (R 4.3.0)
 tidyselect       1.2.0      2022-10-10 [1] RSPM (R 4.3.0)
 tidyverse      * 2.0.0      2023-02-22 [1] RSPM (R 4.3.0)
 timechange       0.2.0      2023-01-11 [1] RSPM (R 4.3.0)
 tzdb             0.4.0      2023-05-12 [1] RSPM (R 4.3.0)
 utf8             1.2.4      2023-10-22 [1] RSPM (R 4.3.0)
 V8               4.4.0      2023-10-09 [1] RSPM (R 4.3.0)
 vctrs            0.6.4      2023-10-12 [1] RSPM (R 4.3.0)
 vroom            1.6.4      2023-10-02 [1] RSPM (R 4.3.0)
 withr            2.5.1      2023-09-26 [1] RSPM (R 4.3.0)
 xfun             0.40       2023-08-09 [1] RSPM (R 4.3.0)
 xtable           1.8-4      2019-04-21 [1] RSPM (R 4.3.0)
 xts              0.13.1     2023-04-16 [1] RSPM (R 4.3.0)
 yaml             2.3.7      2023-01-23 [1] RSPM (R 4.3.0)
 zoo              1.8-12     2023-04-13 [1] RSPM (R 4.3.0)

 [1] /usr/local/lib/R/site-library
 [2] /usr/local/lib/R/library

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Code

options(width = 80L)